AQ 232: Fish Population Dynamics and Stock Assessment
Fish Population Dynamics
Nyamisi Peter
2026-01-21
Mortality (M)
Mortality (M):– The deaths in the population from all causes (natural mortality like disease/predation, plus fishing mortality)
Types of Mortality:
Natural mortality (M) - Disease, predation, starvation, old age
Fishing mortality (F) - Deaths from fishing
Total mortality (Z) - Sum of natural and fishing mortality
Z = M + F
Mortality (M)…
Mortality characteristics:
Often higher at young ages
May increase with old age
Fishing mortality controllable, natural not
Varies by season and location
Mortality Illustration
graph TD
A["Starting Population<br/>1000 fish"] --> B["Natural Mortality<br/>50 fish die"]
A --> C["Fishing Mortality<br/>200 fish caught"]
B --> D["Survivors<br/>750 fish"]
C --> D
D --> E["Total Mortality = 25%"]
Population Models
Mathematical Tools for Prediction
Model Basics
What is a Population Model?
Mathematical representation of how population changes
Uses vital rates (recruitment, growth, mortality)
Predicts future population size
Basis for harvest decisions
Model Basics…
Simple Population Equation:
\[N_{t+1} = N_t + R_t - M_t - C_t\]
Where:
\(N_t\) = population at time t
\(R_t\) = recruitment
\(M_t\) = natural mortality
\(C_t\) = catch
Exponential Growth Model
The Exponential Growth Model describes population growth when there are no resource limitations.
It assumes that the population grows at a constant rate, leading to exponential increase over time.
Formula:
\[N_t = N_0 e^{rt}\]
Or equivalently:
\[N_t = N_0 \lambda^t\]
Where:
\(N_t\) = population size at time t
\(N_0\) = initial population size
\(r\) = intrinsic rate of increase (growth rate)
\(\lambda\) = population growth rate
\(t\) = time
What it means:
If \(r > 0\): population grows exponentially
If \(r = 0\): population stable
If \(r < 0\): population declines or shrink
Exponential Growth Model
Assumption:
1. Unlimited resources
Food is always available
Space is not limiting
No environmental constraints
Habitat quality doesn’t change
Exponential growth curve:
Exponential Growth Model
Assumption:
2. Constant Growth Rate (r)
Growth rate doesn’t change over time
Same regardless of population size
Environmental conditions stable
No density-dependent effects
Exponential growth curve:
Exponential Growth Model
Assumption:
3. No Density Dependence
Population size doesn’t affect vital rates
Birth rate stays constant regardless of how many fish there are
Death rate stays constant
Competition doesn’t increase with density
Exponential growth curve:
Exponential Growth Model
Assumption:
4. Closed Population
No immigration (fish moving in)
No emigration (fish moving out)
All changes come from births and deaths only
Exponential growth curve:
Exponential Growth Model
Assumption:
5. Continuous Population
Applies to every individual equally
No age/size structure effects
Simplified view of population
Exponential growth curve:
Exponential Growth Model
Assumption:
6. No Fishing or External Removal
All removals are natural deaths only
Assumes no harvesting
Exponential growth curve:
Why these assumptions are unrealistic?
Problem with Exponential Model
Ignores resource limits
Fish DO have limits (food, space, oxygen)
Predicts infinite growth
Growth rates DO change with density
Competition DOES increase when crowded
Real populations have age/size structure
Unrealistic
Fisheries involve harvesting
Logistic Growth Model
Logistic growth model is more realistic for fish populations than the exponential growth model.
It incorporates the concept of carrying capacity (K), which represents the maximum population size that the environment can sustain indefinitely and density-dependent effects.
Logistic Growth Model…
Assumption: Growth limited by carrying capacity (K)
\(b\) = density-dependent parameter (strength of density dependence)
Beverton-Holt Model…
Characteristics:
Recruitment increases with spawners
Recruitment plateaus at high spawning stock (asymptotic)
Maximum recruitment = \(a/b\)
Good for species with strong density-dependent effects
How Beverton-Holt Works
At low spawning stock (S_t small):
Recruitment increases nearly linearly with spawners
Few density-dependent effects
Each new spawner adds substantially to recruitment
At high spawning stock (S_t large):
Recruitment plateaus (approaches maximum)
Density-dependent effects dominate
Competition for food/space reduces per-capita recruitment
Additional spawners add little to total recruitment
Mathematical insight of Beverton-Holt model
Formula:
\[R_t = \frac{aS_t}{1 + bS_t}\]
When \(S_t = 0\): \(R_t = 0\) (no spawners = no recruitment)
When \(S_t\) → ∞: \(R_t\) → \(a/b\) (maximum recruitment)
Ricker Model
Formula:
\[R_t = aS_t e^{-bS_t}\]
Where:
\(a\) = productivity parameter
\(b\) = density-dependent parameter
Ricker Model…
Characteristics:
Recruitment increases then decreases
Peak recruitment at intermediate spawning stock
Decline at very high spawner numbers
Good for species (salmon) with cannibalism or crowding effects
Beverton-Holt vs. Ricker
Key Difference:
Beverton-Holt: More spawners = more recruitment (asymptotic)
Ricker: Too many spawners can reduce recruitment (hump-shaped)
Variability in Recruitment
graph TD
A["Spawning Stock Size"] --> B["Environmental<br/>Variation"]
A --> C["Larval Food<br/>Availability"]
A --> D["Temperature<br/>& Currents"]
A --> E["Predation<br/>Pressure"]
B --> F["Recruitment<br/>Uncertainty"]
C --> F
D --> F
E --> F
Practical Implication:
Can’t predict recruitment precisely
Must account for uncertainty
Risk of recruitment failure
Density-Dependent Effects
How Population Density Matters
What is Density Dependence?
Density Dependence:– When vital rates (growth, survival, reproduction) depend on population density
How it works:
High density → increased competition
Competition for food, space, mates
Reduced growth and survival
Lower recruitment
Density Dependence…
Equation:
\[\text{Vital Rate} = f(\text{Population Size})\]
Meaning: Vital Rate is a function of Population Size
The vital rate changes depends on how many fish are in the population
It is a mathematical function where population size is the input
Density Dependence…
Example 1: Growth Rate
\[Growth~Rate = f(Population~Size)\]
Small population: Fish have plenty of food → Fast growth
Large population: Competition for food → Slow growth
Same species, different growth rate based on population density
Density Dependence…
Example 2: Survival Rate
\[Survival~Rate = f(Population~Size)\]
Small population: Low mortality (abundant resources) → High survival
Large population: High mortality (starvation, disease) → Low survival
Density Dependence…
Example 3: Reproduction Rate
\[Reproduction~Rate = f(Population~Size)\]
Small population: Fish produce many eggs → High recruitment
Large population: Fish produce fewer eggs → Low recruitment